Metamath Proof Explorer


Theorem lcvntr

Description: The covers relation is not transitive. ( cvntr analog.) (Contributed by NM, 10-Jan-2015)

Ref Expression
Hypotheses lcvnbtwn.s 𝑆 = ( LSubSp ‘ 𝑊 )
lcvnbtwn.c 𝐶 = ( ⋖L𝑊 )
lcvnbtwn.w ( 𝜑𝑊𝑋 )
lcvnbtwn.r ( 𝜑𝑅𝑆 )
lcvnbtwn.t ( 𝜑𝑇𝑆 )
lcvnbtwn.u ( 𝜑𝑈𝑆 )
lcvnbtwn.d ( 𝜑𝑅 𝐶 𝑇 )
lcvntr.p ( 𝜑𝑇 𝐶 𝑈 )
Assertion lcvntr ( 𝜑 → ¬ 𝑅 𝐶 𝑈 )

Proof

Step Hyp Ref Expression
1 lcvnbtwn.s 𝑆 = ( LSubSp ‘ 𝑊 )
2 lcvnbtwn.c 𝐶 = ( ⋖L𝑊 )
3 lcvnbtwn.w ( 𝜑𝑊𝑋 )
4 lcvnbtwn.r ( 𝜑𝑅𝑆 )
5 lcvnbtwn.t ( 𝜑𝑇𝑆 )
6 lcvnbtwn.u ( 𝜑𝑈𝑆 )
7 lcvnbtwn.d ( 𝜑𝑅 𝐶 𝑇 )
8 lcvntr.p ( 𝜑𝑇 𝐶 𝑈 )
9 1 2 3 4 5 7 lcvpss ( 𝜑𝑅𝑇 )
10 1 2 3 5 6 8 lcvpss ( 𝜑𝑇𝑈 )
11 9 10 jca ( 𝜑 → ( 𝑅𝑇𝑇𝑈 ) )
12 3 adantr ( ( 𝜑𝑅 𝐶 𝑈 ) → 𝑊𝑋 )
13 4 adantr ( ( 𝜑𝑅 𝐶 𝑈 ) → 𝑅𝑆 )
14 6 adantr ( ( 𝜑𝑅 𝐶 𝑈 ) → 𝑈𝑆 )
15 5 adantr ( ( 𝜑𝑅 𝐶 𝑈 ) → 𝑇𝑆 )
16 simpr ( ( 𝜑𝑅 𝐶 𝑈 ) → 𝑅 𝐶 𝑈 )
17 1 2 12 13 14 15 16 lcvnbtwn ( ( 𝜑𝑅 𝐶 𝑈 ) → ¬ ( 𝑅𝑇𝑇𝑈 ) )
18 17 ex ( 𝜑 → ( 𝑅 𝐶 𝑈 → ¬ ( 𝑅𝑇𝑇𝑈 ) ) )
19 11 18 mt2d ( 𝜑 → ¬ 𝑅 𝐶 𝑈 )